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The inverse spectral problem for indefinite strings

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journal contribution
posted on 04.10.2018 by Jonathan Eckhardt, Aleksey Kostenko
Motivated by the study of certain nonlinear wave equations (in particular, the Camassa–Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form −u" = z u ω + z2u υ on an interval [0, L), where ω is a real-valued distribution in H −1loc [0, L), υ is a non-negative Borel measure on [0, L) and z is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein’s celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings.

Funding

Research supported by the Austrian Science Fund (FWF) under Grants No. J3455 and P26060.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Inventiones mathematicae

Volume

204

Issue

3

Pages

939 - 977

Citation

ECKHARDT, J. and KOSTENKO, A., 2016. The inverse spectral problem for indefinite strings. Inventiones mathematicae, 204(3), pp. 939-977.

Publisher

© Springer

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2016

Notes

This is a post-peer-review, pre-copyedit version of an article published in Inventiones mathematicae. The final authenticated version is available online at: https://doi.org/10.1007/s00222-015-0629-1

ISSN

0020-9910

eISSN

1432-1297

Language

en

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